Question

Find the distance between the point (-3,-4), and the line 2y =-3x+6

Answer 1

do you know how to find a perpendicular line?

Answer 2

NO

Answer 3

If you have a line with a slope of -3/2 then the slope of a line perpendicular to it will have a negative reciprocal of it, that is 2/3. But you want this perpendicular line to go through the point (-3,4) so we need to use the new perpendicular equation y= (2/3) x + b to find the y-intercept, b:

$$
4=(2/3)(-3)+b
$$
From which you can now get b.

This perpendicular line intersects the original line when both of these equations are equal
$$
(2/3)x+b=(-3/2)x+6
$$
Solve for x to find where on the x-axis these lines intersect and then plug in to either equation to find y. This gives you where the two points intersect.

But why do we want to find this perpendicular line that goes through the given point? We do this because the shortest distance between a point and a line is THE line that is perpendicular to the line and goes through the given point like this:



Once you have this point, where the perpendicular intersects the original line. Use the coordinates to find the distance, d:

$$
d=\sqrt{(x_1-x_2)^2+(y_1+y_2)^2}
$$
Where \(x_1=-3,y_1=-4\), the point given, that is (-3,-4), and \((x_2,y_2)\) is the point where the perpendicular line intersects the original line.

Doe this make sense?